Polylogarithmic Approximation Algorithms for Weighted-ℱ-deletion Problems

Abstract

For a family of graphs ℱ, the W<scp;>eighted</scp;> ℱ V<scp;>ertex</scp;> D<scp;>eletion</scp;> problem, is defined as follows: given an n -vertex undirected graph G and a weight function w : V ( G )࢐ ℝ, find a minimum weight subset S ⊆ V ( G ) such that G - S belongs to ℱ. We devise a recursive scheme to obtain O(log O(1) n )-approximation algorithms for such problems, building upon the classical technique of finding balanced separators . We obtain the first O(log O(1) n )-approximation algorithms for the following problems. • Let F be a finite set of graphs containing a planar graph, and ℱ= G ( F ) be the maximal family of graphs such that every graph H ∈ G ( F ) excludes all graphs in F as minors. The vertex deletion problem corresponding to ℱ= G ( F ) is the W eighted P lanar F -M inor -F ree D eletion (WP F -MFD) problem. We give a randomized and a deterministic approximation algorithms for WP F -MFD with ratios O(log 1.5 n ) and O(log 2 n ), respectively. Prior to our work, a randomized constant factor approximation algorithm for the unweighted version was known [FOCS 2012]. After our work, a deterministic constant factor approximation algorithm for the unweighted version was also obtained [SODA 2019]. • We give an O(log 2 n )-factor approximation algorithm for W eighted C hordal V ertex D eletion , the vertex deletion problem to the family of chordal graphs. On the way to this algorithm, we also obtain a constant factor approximation algorithm for M ulticut on chordal graphs. • We give an O(log 3 n )-factor approximation algorithm for W eighted D istance H ereditary V ertex D eletion . We believe that our recursive scheme can be applied to obtain O(log O(1) n )-approximation algorithms for many other problems as well.